The Numerical Initial Boundary Value Problem for the Generalised Conformal Field Equations in General Relativity

Abstract

The purpose of this work is to develop for the first time a general framework for the Initial Boundary Value Problem (IBVP) of the Generalised Conformal Field Equations (GCFE). At present the only investigation toward obtaining such a framework was given in the mid 90's by Friedrich at an analytical level and is only valid for Anti-de Sitter space-time. There have so far been no numerical explorations into the validity of building such a framework.

The GCFE system is derived in the space-spinor formalism and Newman and Penrose's eth-calculus is imposed to obtain proper spin-weighted equations. These are then rigorously tested both analytically and numerically to confirm their correctness. The global structure of the Schwarzschild, Schwarzschild-de Sitter and Schwarzschild-Anti-de Sitter space-times are numerically reproduced from an IVP and for the first time, numerical simulations that incorporate both the singularity and the conformal boundary are presented.

A framework for the IBVP is then given, where the boundaries are chosen as arbitrary time-like conformal geodesics and where the constraints propagate on (at least) the numerical level. The full generality of the framework is verified numerically for gravitational perturbations of Minkowski and Schwarzschild space-times. A spin-frame adapted to the geometry of future null infinity is developed and the expressions for the Bondi-mass and the Bondi-time given by Penrose and Rindler are generalised. The Bondi-mass is found to equate to the Schwarzschild-mass for the standard Schwarzschild space-time and the famous Bondi-Sachs mass loss is reproduced for the gravitationally perturbed case.